Similar bounds on the local treewidth of random regular graphs \(G(n,d)\)  can be established via similar arguments to those used for \(G(n,d/n).\) 
For the upper bound, one can use the fact [1]} that for every \(k<nd/4\)  distinct unordered pairs of vertices, the probability they all occur simultaneously in \(G(n,d)\) 
is at most \((2d/n)^k\)  and then nearly identical arguments to those in Theorem REF . For the lower bound one can use the fact that w.h.p. \(G(n,d)\)  is a \((1/2,\Omega (d))\)  expander [2]}, [3]} from which it easily follows that it is also a \((2/3,\Omega (d)\) ) expander
(for sets of size larger than \(n/2\)  count the edge coming out of the complementary set) and afterwards use Theorem REF .
We summarize this with the following Theorem:
